Transforming Cartesian to Polar Coordinates

shanepitts
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Homework Statement


I am currently trying to calculate the moment and products of inertia of a ring rotating about the x-axis at the moment the ring lies in the xy plane. The problem is that the notations I have from textbook are denoted for Cartesian coordinates. i.e. Ixx=∑i mi(yi2+zi2), and Ixy=∑imixiyi. How can I convert these to polar coordinates?
image.jpeg


Homework Equations


x=rcosθ, y=rsinθ, z=?

The Attempt at a Solution


I'd assume that the moment of inertia would become Ixx=∑mr2dθ.

but since the problem is for a ring with linear mass density, I am also wondering must I exclude certain coordinates?

thank you in advance.
 
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You can find descriptions of how to calculate such moments of inertia at http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html
Be careful that for continuous objects, not point masses, you have to use density instead of masses and integrals instead of sums.

For symmetry reasons, you have that Ixy = 0, Ixz = 0, etc., and also Ixx = Iyy.
 
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